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fluid mechanics explained

what is Reynolds number

Reynolds number Re = ρvD/μ (inertia/viscosity); critical Re~2300 splits laminar from turbulent; dimensionless

The same water sometimes flows smooth and sometimes tangled. What splits the two? Drag the Re slider and see the flow change. One number splits smoothness from tangle.

Set one small cylinder in the flow and drag Re, and the difference is clear. When Re is low the water turns smoothly around the cylinder and the wake is calm. This orderly layered flow is laminar. Raise Re and vortices peel off one after another behind the cylinder and tangle, and that is turbulent. The same cylinder has two faces.

Laminar or turbulent is set by a tug of war of two forces. Inertia pushes a disturbance onward and grows it, and viscosity smooths it and settles it, and Re is exactly the ratio of this inertia divided by viscosity. Drag Re and you see which side wins. When inertia wins it is turbulent. When viscosity wins it is laminar.

The switch is sudden. Raise Re slowly and the moment it crosses a threshold the smooth flow starts to break up, and in pipe flow that boundary is around Re 2300. Below it is laminar, above it is turbulent, and between them is a transition region where the two waver. The moment you cross the boundary the face changes.

In symbols, Re = ρvDμ, a dimensionless number, density times speed times size divided by viscosity. Drag v and Re follows and crosses the threshold line. The faster, the thicker, and the thinner the viscosity, the larger Re grows and the flow leans turbulent. It is a pure number with no unit, so a river, a blood vessel, and an oil pipeline can be compared on the same measure.

A faucet is an easy picture. Open the water a little and a smooth clear stem falls like a glass rod, and that is laminar. Drag the flow rate and open it harder and at some moment the stem bubbles and breaks up white and crosses into turbulent. Incense smoke that rises straight at first and splits above is the same. How viscosity shapes the velocity distribution inside a pipe is taken up by the laminar parabola.

In PracticeIn short, whether a flow is laminar or turbulent is set by the tug of war of inertia and viscosity, and that ratio is the Reynolds number Re = ρvDμ. When Re is small viscosity wins and the flow is a smooth laminar one, and when it is large inertia wins and the flow is a tangled turbulent one, and the critical Re of pipe flow is about 2300. The faster, the thicker, and the thinner the viscosity, the larger Re grows. A dimensionless number with no unit lets flows of wildly different size and speed be compared on one measure, and even makes a scaled model experiment possible, and that is the seed of dimensional analysis. The parabolic velocity distribution that viscosity shapes inside a pipe follows.
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